What you mention about having to choke a EZ is normal even when you get it back to normal.
Most generally when cold you have to set the comp release to on, then full choke until it pops, then 1/2 choke for start and then let it warm up on 1/2 choke. If it dies when turning off 1/2 choke and the engine is still cold you have to repeat the full choke routine. (it's just the nature of the little beast.
When hot set to full choke, pull one time, then go to 1/2 choke for a start then off for run. You can usually take it off 1/2 choke sooner when the engine is hot or warm.
(they like a choke and very rarely do they flood and most generally always have to do at least one full choke even when hot.
If the throttle lock fast idle adjustment screw is set really low some EZ's are harder to start cold and hot. (I'm referring to the fast idle throttle lock screw that is located in the front of the handle.
You need to check the in tank fuel filter, the gas line, the little fine screen in the HDC. If that does not do it, get a kit for the HDC. I suspect the pump side of the HDC. this is the side that has the little flapper valves, but with a carb that old it could be
both the pump and the diaphragm side. Keep a heads up on the kit, sometimes the new kit has a needle valve that is little bit different size than the existing needle and require the metering lever to require a slight re-tweaking of it's height.
You can go over the the Walbro site and download service info for that carb. Pay close attention to the sandwiching of the gaskets layout.
Them EZ's are built for the long haul, really good little saws and worth the effort to get it correct.
Usually what does them in is long term storage without starting the engine and use/storage of ethanol fuel.
I've got some good running EZ's that are over 40 years old and used in wood lot for trimming every year. One 47year old EZ only required one carb kit and few replacement chains, still has same bar. Very reliable little saws and powerful for their size.